p-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

Let (xₙ)n≥0 be a linear recurrence sequence of order k≥2 satisfying xₙ=a₁xₙ₋₁+a₂xₙ₋₂++aₖxₙ₋ₖ for all integers n≥ k, where a₁,,aₖ,x₀,, xₖ₋₁∈ Z, with aₖ≠0. In 2017, Sanna posed an open question to classify primes p for which the quotient set of (xₙ)n≥0 is dense in Qₚ. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root ± α, where α is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in Qₚ, then the quotient set of (xₙ)n≥ 0 is dense in Qₚ. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over Q.